Cardinality

Definition

Let $A$ be a set. The cardinality of $A$, written $|A|$, is an object assigned to $A$ such that, for all sets $A$ and $B$, $$|A| = |B| \iff A \text{ and } B \text{ are } \text{[equinumerous](m2node:08EJHHMB5R)}.$$ For a finite set $A$, $|A|$ is the number $n$ of its elements, that is, the unique $n \in \mathbb{N}$ with $A \approx \{1, \dots, n\}$. For an infinite set, $|A|$ is a cardinal number.

Remarks

Other notations are $\#A$, $\operatorname{card}(A)$ and $\overline{\overline{A}}$. With the axiom of choice, one can take $|A|$ to be the least ordinal equinumerous with $A$. $|A| \le |B|$ means that there is an injection from $A$ to $B$. See order of cardinals.

Explanations

L1 human

The cardinality of $\{a, b, c\}$ is $3$. For infinite sets, cardinality is a new kind of number that records which sets can be put in bijection. For example, $|\mathbb{N}| = |\mathbb{Z}|$, but $|\mathbb{N}| < |\mathbb{R}|$.